Derivative of x√x: Multiple Approaches for Calculus

The derivative of a function is a fundamental concept in calculus, representing the rate of change of the function. In this article, we explore the derivative of the function fx x √x using both the power rule and the limit definition of the derivative. We will provide detailed derivations and explanations to help you understand the process from different angles.

1. Using the Power Rule

The power rule is a useful tool for differentiating functions of the form fx x^n. It states that fx nx^(n-1). Let us apply this rule to the function fx x √x.

Step 1: Rewrite the function in the form of a power. x √x can be written as x * x^(1/2) or x^(3/2).

Step 2: Apply the power rule. Let n 3/2, then fx (3/2) * x^(3/2 - 1) (3/2) * x^(1/2).

Step 3: Simplify the expression. Therefore, fx (3/2) * x^(1/2) (3/2) * √x.

This approach is straightforward and efficient, leveraging the power of the power rule to quickly find the derivative.

2. Using the Power Rule Further

To illustrate the power rule in more detail, let us break down the steps of differentiating fx x * x^(1/2).

Step 1: Differentiate the first term, x. dx/dx 1.

Step 2: Differentiate the second term, x^(1/2) using the power rule. 1/2 * x^(-1/2) 1/2 * 1/√x.

Step 3: Combine the results. 1 * x^(1/2) x * (1/2 * x^(-1/2)) √x (1/2) * (x / √x) √x (1/2) * √x (3/2) * √x.

This provides a detailed breakdown of the power rule application, helping to understand each component of the derivative process.

3. Using the Limit Definition of the Derivative

The limit definition of the derivative provides an alternative and more fundamental approach to finding derivatives. The definition is given by:

f'x lim(h→0) (f(x h) - f(x)) / h

Let us apply this to fx x √x.

Step 1: Express fx in terms of x and x h. fx x √x and f(x h) (x h) √(x h).

Step 2: Formulate the difference quotient. (f(x h) - f(x)) / h ((x h) √(x h) - x √x) / h.

Step 3: Factor and simplify. First, multiply by the conjugate to simplify the expression.

((x h) √(x h) - x √x) * ( (x h) √(x h) x √x) / h ( (x h) √(x h) x √x). Expand the numerator and simplify.

Step 4: Take the limit as h approaches 0. lim(h→0) ( (x h) √(x h) - x √x) / h lim(h→0) ( (x h) √(x h) - x √x) / h 1/2 √x.

Combining these steps, we obtain the derivative as 1 (1/2) * √x (3/2) * √x.

Conclusion

Through these multiple methods, we have demonstrated how to find the derivative of the function fx x √x. This approach not only reinforces the power rule but also deepens the understanding of the limit definition of the derivative.

Keywords: derivative of x √x, power rule, limit definition of derivative